Tuesday, September 18, 2007

This Time, You Ask the Questions

Was idly talking about logic puzzles and recalled a variation on the familiar Knights and Knaves. So anyway:(Updated with more possible deities)

You live on an island ruled by three deities who will answer yes/no questions for the inhabitants. They're overly obsessed with never appearing in the same form twice, so you never know which one is which when you show up at the temple. Truth always tells the truth. Liar always lies. Random Whimsy will switch between telling the truth and lying whenever he darn well pleases, and will favor whichever aids him personally. All three deities travel together and answer all questions, so you can't direct a question to a specific deity.

Since you're tired of chronically playing games of Knights and Knaves every time you've got a trivia question you want answered, you decide to thin down the numbers: The ancient scrolls or some such say that if a deity is unable to produce a yes/no answer due to a paradox, his head will explode.

So, what questions can you ask to identify each deity (fairly easy, no points), kill a specific deity which will not harm the other two (2 points), or a question that'll kill two deities without harming another (5 points). Minus infinity points for a question that'll kill all three, since that'll arouse the attention of the Hounds of Tindalos, who'll rip you to shreds, or whatever they're supposed to do. All I know is that I like their plush.

Alternate deities to replace Random/Whimsy, to possibly expand the challenge:

Chaos: Switches randomly between honesty and lying.Alternator: Switches between telling the truth and lying with each question. First answer is random.Yes-Man: If Truth and Liar give the same answer, he will give the same answer as well. If Truth and Liar give different answers, he will lie.Ragnarok: If Truth's and Liar's heads explode, he will answer to cause his head to explode. If that option isn't available, he tells the truth.

12 comments:

Alright, let's start out with trying to identify them. Random is going to be tough to pick out alone, but you can count on Truth and Liar to give at least deterministic answers so you've got a chance there.

My first goal would be to pick out Random so that he won't bog down the results. To do this, I would start out by asking a series of questions to which I already know the answer, such as "Does 2 + 2 = 4?" Truth and Liar will give opposite answers, and Random will be the same as one of those. Then, you keep at it until the two who were paired in previous questions split. The one who's switched to agree with the third is Random. Of course, the other two are now going to be no problem since you've gotten either the truth or lies from them on past questions.

Now, for some deity killing. Killing Liar alone is easy, as we resort to the simple liar's paradox: "Are you lying?" Liar is bound to lie, but is either an answer of "yes" or "no" a lie? If he says "yes," this resolves into the liar's paradox and he's lying that he's lying so he's telling the truth. But if he says "no," then he's either lying - so he means "yes" - or he's telling the truth, but he's incapable of telling the truth, so he must be lying. Poof, Liar's dead.

Now, Truth and Random are going to be immune to this one. Truth simply says "no," and he's telling the truth that he's not lying. Random also simply chooses to tell the truth here and say "no," so he survives (he wouldn't want to die, would he?).

Next thing to try: Killing Truth. He's the other deterministic one so it should be possible. As before, self-reference is our friend, so let's try the question: "What answer are you not giving me to this question?" There's no truthful answer to this question, as whatever answer you give is the opposite of the answer you aren't giving. Liar and Random are saved, however, as they can simply lie and say anything.

I don't want to take the fun away from anyone else who comes by, so I'll leave it at that for now, while I puzzle out a way to kill just Random or two deities at once.

Not that it's about the points, but I believe your post just said 2 points for each question that kills a specific deity.

But anyways, I suspect that it might actually be possible to kill Random alone if you can figure out a question that cannot be answered as "Yes" or "No" as either a lie or the truth, and then package it on the condition that it only works this way if Random is the one answering. Then again, this might not work under the revised rules.

Okay, I figured out how to kill off just Random/Whimsy. Ask the following question: "Is the following statement true: 'You are Random/Whimsy and this statement is false.'?"

The first part of this statement is false for both Truth and Liar, so Truth can safely say "No" and Liar can safely say "Yes," regardless of the truth value of the second part of the statement.

For Random, the first part is going to be true regardless of what he says or whether he chooses to lie or tell the truth. Since it's connected with an "and," we can then take it out of consideration and his answer will only apply to the second half of the statement.

So we've essentially just asked Random the question: "Is the following statement true: 'This statement is false.'?" Whether he's lying or telling the truth in his answer, he's giving a possible solution to the Liar's Paradox. Since the statement is neither true nor false, either answer leads to a paradox. Thus, Random's head explodes.

Actually, this question format is very powerful, as changing only the names in the first part lets us target whoever we want. If we change the question to, "Is the following statement true: 'You are Random/Whimsy or Truth and this statement is false.'?" then we kill both Random/Whimsy and Truth. To kill Truth and Liar, we change it to "Is the following statement true: 'You are Truth or Liar and this statement is false.'?" And finally, to kill Liar and Random/Whimsy, we change it to "Is the following statement true: 'You are Random/Whimsy or Liar and this statement is false.'?"

The last one would probably be the best choice, as you're left with only the truth-telling god alive, and all your questions from now on will be answered truthfully.

Info's liar's paradox does work! If you ask Liar "Are you lying?" and he says no then he's fine because he is actually lying (because he always lies), so his statement 'no' is false, which doesn't hurt him any because he always makes false statements. So he's fine. A liar can claim to tell the truth without his head exploding.

The only way you can really get out of that particular question is to say "no" and have it be the truth, equating to saying "I am not lying" and it being a true statement. If you say "no" and have it be a lie, then you're implying that "I am not lying" is a false statement, which means you're telling the truth when you say you're not lying. But didn't we just establish that he wasn't telling the truth? Hence we have a logical contradiction.

Besides that, the question, "Is the following statement true: 'You are Liar and this statement is false.'?" should be able to take him down anyways.

Hmm, well I think my question format works for when he's there as well. For instance, ask the question of the three: "Is the following statement true: 'You are Truth or Liar and this statement is false.'?" Now, to Ragnarok, the statement is simply false. He can choose to lie and say "Yes" or tell the truth and say "No," but neither will make his head explode, so he'll just tell the truth.

Heh, don't worry though, I've actually got a good idea of something else to do with this puzzle. You should see what I'm up to in a few weeks. .>